# Poisson Distribution of sum of two random independent variables XX, YY

$X \sim \mathcal{P}( \lambda)$ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim \mathcal{P}( \lambda + \mu)$ but I don’t understand how to derive it.

## Answer

This only holds if $X$ and $Y$ are independent, so we suppose this from now on. We have for $k \ge 0$:

Hence, $X+ Y \sim \mathcal P(\mu + \lambda)$.

Attribution
Source : Link , Question Author : Community , Answer Author : martini